Classification of Surfaces

The classification of closed connected surfaces is one of the problems which gave
rise to modern topology, and will be the highlight of Math. 655. Recall that a
closed surface is a space which in the vicinity of each point looks like the plane
and which satisfies the finiteness condition: the surface can be cut up into a
finite number of pieces each homeorphic to a disk. (To simplify the
classification problem we are avoiding consideration of surfaces with boundary,
like a cylinder or Möbius strip.)

There is a very nice answer to this problem. First of all, there are the orientable
surfaces:

These consist of the sphere and finite connected sums of tori. The connected sum
construction connects two surfaces with a tube (after cutting out holes in the
surfaces where the tubes are attached).

More problematic are the unorientable surfaces. To cut a long story short, all the closed
connected unorientable surfaces can be constructed by cutting out a finite number of holes from
a sphere and then abstractly gluing in Möbius strips along the boundary of each
hole. Note that the boundary of a Möbius strip is a simple closed curve, and thus
can be mated (abstractly) to the boundary of a hole. Unfortunately the boundary of
the Möbius strip is a knotted simple closed curve. (For the standard picture of
the Möbius strip the boundary is a trefoil knot.) Thus it is difficult to
visualize this construction.

While it is impossible to faithfully represent closed unorientable surfaces as surfaces
in 3-dimensional Euclidean space, it is possible to do so if we allow selfintersections,
and this gives us some idea of what the surface looks like. (We have already seen this
with the Klein bottle.)

First we need a new representation of the Möbius strip:

This representation of the Möbius strip is called the crosscap.
Unlike the usual representation of the Möbius strip, it has a selfintersection:
the seam of the crosscap where the surface selfintersects is not a line, as it
appears in the picture, but actually a simple closed curve. The following figure
shows how to construct a cardboard model of the crosscap:

and the following animation shows what the result looks like:

The line of selfintersection between the green and red triangles in the center
consists of double points, except for the two endpoints. Thus in the actual
crosscap (ie. the Möbius strip) there are two distinct edges, one in the
green triangle and one in the red triangle, and the two edges are glued to each
other at the endpoints to form a simple closed curve. The following abstract
gluing picture of the crosscap

may help you to see that it is homeomorphic to the Möbius strip. Note that
the diagonally opposite corners of the inner square are glued together, and that
the two edges referred to above are a and b. To see the correspondence
between this picture and the 3-dimensional one shown in the animation, pull the points
which are the heads of the a arrows (and tails of the b arrows) up
out of the original plane of the figure, while keeping the points which are the tails
of the a arrows (and heads of the b arrows) in the original plane.
Then bring the two pairs of points together, being careful to leave the second pair
within the original plane.

The one advantage that the crosscap has over the more usual nonselfintersecting
representation of the Möbius strip in 3-dimensional space is that its boundary
is unknotted. Hence crosscaps can be physically glued into holes in a sphere, thus
yielding (selfintersecting) pictures of the closed unorientable surfaces in
3-dimensional space:

The first surface shown above is called the real projective plane,
and the second can be shown to be an alternative representation of the Klein bottle.

We can now state the classification theorem for surfaces precisely:

Classification Theorem for SurfacesAny closed connected
surface is homeomorphic to exactly one of the following surfaces: a sphere, a finite
connected sum of tori, or a sphere with a finite number of disjoint discs removed and
with crosscaps glued in their place. The sphere and connected sums of tori are
orientable surfaces, whereas surfaces with crosscaps are unorientable.

Möbius was the first to attempt the classification of
surfaces. In an 1870 paper he proved the above theorem for orientable surfaces
smoothly imbedded in 3-dimensional Euclidean space. The classification of
unorientable surfaces was first announced by
W. von Dyck in 1888, but his proof was
incomplete. (Among other problems, at that time there was no satisfactory concept
of an abstract surface, not imbedded in Euclidean space.) The first essentially
rigorous proof of the classification theorem was given by
M. Dehn and P. Heegard in 1907,
under the assumption that surfaces can be triangulated (ie. cut up into a finite
number of (curved) triangles intersecting with each other along (curved) edges or
vertices.) The triangulability of surfaces was first proved by
T. Rado in 1925,
thus completing the proof of the classification theorem.

A key ingredient in the proof of Rado's theorem is (a strong
form of) the Jordan curve theorem:
any simple closed curve in the plane separates the plane into two regions, which
was proved by
A. Schönflies in 1906. (His proof contained some errors which were fixed by
L E J Brouwer in 1909.) It is easy to see that the
Jordan curve theorem for the plane is equivalent to the Jordan curve theorem for
the sphere:

The Jordan curve theorem is not true for the other closed surfaces. There are simple
closed surfaces on these surfaces which do not separate the surface. The appropriate
generalization of the Jordan curve theorem for arbitrary closed surfaces is given
below. It is stated in terms of genus of a surface, a concept which we
define as follows:

A sphere is defined to have genus 0, the connected sum of g toris is defined
to have genus g and an unorientable surface with g crosscaps is
defined to have genus g-1.

Jordan Curve Theorem for SurfacesThe maximum number of disjoint
simple closed curves which can be cut from an orientable surface of genus g
without disconnecting it is g. The maximum number of disjoint simple closed
curves which can be cut from an unorientable surface of genus g
without disconnecting it is g+1.

The following pictures show maximal nonseparating systems of disjoint simple closed
curves on the connected sum of two tori (genus 2):